Jan Reiterman

The Birkhoff theorem for finite algebras

A finite analogue of the Birkhoff variety theorem is proved: a non-void class of finite algebras of a finite type τ is closed under the formation of finite products, subalgebras and homomorphic images if and only if it is definable by equations for implicit operations, that is, roughly speaking, operations which are not necessarily induced by τ-terms but which are compatible with all homomorphisms. It is well-known that explicit operations (those induced by τ-terms) do not suffice for such an equational description. Topological aspects of implicit operations are considered. Various examples are given.

Effective descent maps of topological spaces

Abstract The basic technique in A. Joyals and M. Tierneys work on “An extension of the Galois theory of Grothendieck” is descent theory for morphisms of locales (in a topos). They showed that open surjections are effective descent morphisms in the category of locales. I. Moerdijk gave an axiomatic proof of this result which shows that the same result holds true also in the category Top of topological spaces. G. Janelidze and W. Tholen proved that every locally sectionable map in Top is an effective descent morphism, and that effective descent morphisms are universal quotient maps in Top. In this paper, we give • • bu a complete characterization of effective descent maps in bdTop, • bu an example of a universal quotient map in bdTop which is not an effective descent morphism. This is done by first transfering the problem into a friendlier environment than Top, namely into the topological quasitopos hull of Top, the category of pseudotopological spaces. Here effective descent morphisms are simply quotient maps. Although the notion of effective descent morphism depends on the category, it is possible to reinterpret the pseudotopological characterization in purely topological terms, under extensive use of filter theory.

CATEGORICAL CONSTRUCTIONS OF FREE ALGEBRAS, COLIMITS, AND COMPLETIONS OF PARTIAL ALGEBRAS

1.1. Given a base category K and a functor F: K -, K, we shall consider the category A(F) of F-algebras and the category PA(F) of partial F-algebras (for definitions, see 1.2). The category A(F) has been studied in a lot of papers [2-7, 13, 171 see also [ll, 121 in connection with the categorical universal algebra and automata and control .theory. We shall deal with the following problems: (1) Existence and construction of free algebras in A(F). (2) Existence and construction of colimits in A(F). (3) Existence and construction of left adjoints to functors A(F) -j A(G) induced by transformations G += F. (4) Completions of partial algebras and other properties of the category PA(F). (5) Cocompleteness of the category of algebras for a triple (F, q, p) in K. The present paper is based on the thesis [15] of the second author which generalizing categorical constructions [l, 4, 7, 131 and classical constructions of universal algebra attempts to form a general theory of constructions of free algebras, colimits etc. Some results of [ 151 are improved and the approach is applied to partial algebras and to algebras for a triple. As a technical tool, we shall embed A(F) into the category A*(F) [15] of algebraized chains which possess convenient properties (Sections 2-4). The category PA(F) will be investigated by means of the embedding into the category GPA(F) of generalized partial algebras. Notice that categories A*(F) and GPA(F) form “completions” of A(F): they have free algebras and are cocomplete. The usefulness

Tree constructions of free continuous algebras

Abstract Continuous algebras are algebras endowed with a partial order which is complete with respect to specified joins and such that the operations preserve these specified joins. We prove the existence of free continuous algebras by actually giving a concrete description of them in terms of trees, for any type of algebras and any choice of the “specified” joins.

Dynamic Algebras which are not Kripke Structures

The first example of a dynamic algebra which is not isomorphic to any Kripke structure was given by Kozen [8]. We analyze properties of dynamic algebras to get general arguments making it possible to construct a lot of other examples.

Cartesian closed hull of the category of uniform spaces

Abstract A concrete category K is a CCT (cartesian closed topological) extension of the category Unif of uniform spaces if 1. K is cartesian closed, 2. Unif is a full, finitely productive subcategory of K and the forgetful functor of K extends that of Unif and 3. K has initial structures. We describe the smallest CCT extension of Unif which is called the CCT hull by H. Herrlich and L.D. Nel. The objects of the CCT hull are bornological uniform spaces, i.e. uniform spaces endowed with a collection of “bounded” sets related naturally to the uniformity; the morphisms are the uniformly continuous maps which preserve the bounded sets.

Dynamic algebras with test

Abstract We generalize some results, known for dynamic algebras, to test algebras. Main results: Every free algebra in the equational class generated by separable test algebras is isomorphic to a Kripke test structure. Consequently, equational classes generated by separable test algebras and by Kripke test structures coincide. In contrast to dynamic algebras, free separable test algebras over finitely many generators do not exist. Epimorphisms in the equational class generated by separable dynamic or test algebras are shown not to be necessarily surjective.

The Birkhoff Variety Theorem for continuous algebras

Varieties of continuous algebras, i.e., classes presented by (in)equalities for terms, are characterized as precisely the HSP classes. Terms here are the usual algebraic terms enriched by iterated join symbols. A discussion of varieties which require only finitely many variables, or only the usual terms, is also presented.

Banach's Fixed-Point Theorem as a base for data-type equations

For a categoryK of data types, solutions of recursive data-type equationsX ℞T(X), whereT is an endofunctor ofK, can be constructed by iteratingT on the unique arrowT1 → 1. This is well-known forK enriched over complete posets and forT locally continuous (an application of the Kleene Fixed-Point Theorem). We prove this forK enriched over complete metric spaces and forT contracting (an application of the Banach Fixed-Point Theorem). Moreover, we prove that each such recursive equation has a unique solution. Our results generalize the approach of P. America and J. Rutten.

Ultrafilters on ω and atoms in the lattice of uniformities II

Abstract Interaction between ultrafilters and uniformities on a countable set is investigated. Various ultrafilters are constructed such that atoms in the lattice of uniformities refining the corresponding ultrafilter uniformities have special properties.